Spring 2022 - MATH 842 G100

Algebraic Number Theory (4)

Class Number: 3312

Delivery Method: In Person

Overview

  • Course Times + Location:

    Jan 10 – Apr 11, 2022: Wed, Fri, 2:30–4:20 p.m.
    Burnaby

Description

CALENDAR DESCRIPTION:

Review of Galois theory, integrality, rings of integers, traces, norms, discriminants, ideals, Dedekind domains, class groups, unit groups, Minkowski theory, ramification, cyclotomic fields, valuations, completions, applications.

COURSE DETAILS:

Algebraic number theory comprises the study of algebraic numbers: numbers that satisfy polynomial equations with rational coefficients. The parallels with usual integer arithmetic are striking, as are the notable differences (as, for instance, failure of unique factorization into prime factors). The subject is fundamental to any further study in number theory or algebraic geometry.

In this course we develop the tools to properly understand unique factorization and its failure. We establish fundamental results such as Dirichlet's Unit theorem and the finiteness of the ideal class group. We highlight the applicability of the algebraic tools we develop to both algebraic numbers and to algebraic curves. Depending on time and interests of the participants, we will also look into various applications and more advanced topics.

Grading

  • Biweekly assignments (weighted equally) 30%
  • Presentation 15%
  • Final Examination (take home) 55%

REQUIREMENTS:

Prerequisite: MATH 440/740

Materials

REQUIRED READING:

Milne, J.S.
Algebraic Number Theory
available from: http://www.jmilne.org/math/CourseNotes/ant.html


RECOMMENDED READING:

Alternative and additional reading:  

Quite algebraic and very careful exposition, including lots of detail in proofs:  
Ribenboim, Paulo
Classical theory of algebraic numbers.
Universitext. Springer-Verlag, New York, 2001.
xxiv+681 pp.
ISBN: 0-387-95070-2  

Very compact in its exposition, but still complete. Probably one of the slickest presentations available:

Neukirch, Jürgen
Algebraic number theory.
Grundlehren der Mathematischen Wissenschaften, 322. Springer-Verlag, Berlin, 1999. xviii+571 pp.
ISBN: 3-540-65399-6  

Quite complete in its coverage of theory but also paying attention to the computational side of things:  

Stein, William
Algebraic Number Theory, a Computational Approach
Full text available from: https://wstein.org/books/ant/

Graduate Studies Notes:

Important dates and deadlines for graduate students are found here: http://www.sfu.ca/dean-gradstudies/current/important_dates/guidelines.html. The deadline to drop a course with a 100% refund is the end of week 2. The deadline to drop with no notation on your transcript is the end of week 3.

Registrar Notes:

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS

SFU’s Academic Integrity web site http://www.sfu.ca/students/academicintegrity.html is filled with information on what is meant by academic dishonesty, where you can find resources to help with your studies and the consequences of cheating.  Check out the site for more information and videos that help explain the issues in plain English.

Each student is responsible for his or her conduct as it affects the University community.  Academic dishonesty, in whatever form, is ultimately destructive of the values of the University. Furthermore, it is unfair and discouraging to the majority of students who pursue their studies honestly. Scholarly integrity is required of all members of the University. http://www.sfu.ca/policies/gazette/student/s10-01.html

TEACHING AT SFU IN SPRING 2022

Teaching at SFU in spring 2022 will involve primarily in-person instruction, with safety plans in place.  Some courses will still be offered through remote methods, and if so, this will be clearly identified in the schedule of classes.  You will also know at enrollment whether remote course components will be “live” (synchronous) or at your own pace (asynchronous).

Enrolling in a course acknowledges that you are able to attend in whatever format is required.  You should not enroll in a course that is in-person if you are not able to return to campus, and should be aware that remote study may entail different modes of learning, interaction with your instructor, and ways of getting feedback on your work than may be the case for in-person classes.

Students with hidden or visible disabilities who may need class or exam accommodations, including in the context of remote learning, are advised to register with the SFU Centre for Accessible Learning (caladmin@sfu.ca or 778-782-3112) as early as possible in order to prepare for the spring 2022 term.